Goren Gordon , Gershon KurizkiWeizmann Institute of Science, Israel

Daniel LidarUniversity of Southern California, USA

QEC07 USC Los Angeles, USADec. 17-21, 2007

OutlineUniversal dynamical decoherence

control formalismBrief overview of

Calculus of VariationsAnalytical derivation of equation

for optimal modulationNumerical resultsConclusions

2 a tR t Fd G

' '

F F K K

y t y y t y

12

1 2 2 20 0

,''

,T t

E t tt

dt dt t t

Z

Z

Decoherence ScenariosIon trap Cold atom in (imperfect)

optical lattice

Ion in cavityKreuter et al. PRL 92 203002 (2004)

Keller et al. Nature 431, 1075 (2004)

Häffner et al. Nature 438 643 (2005)

Jaksch et al. PRL 82, 1975 (1999)Mandel et al. Nature 425, 937 (2003)

Universal dynamical decoherence control formalism

. .

a

j

e

j

j

a

j

H t e e

j j

j e

t

h c

system+modulation

bath

coupling g|

e||

j

j

a

ej

a t

2| R t tF t e t e

Fidelity of an initial excited state:

11

2

1 2*

2 110 0 2

2Re ai t tt t

t etR d t ttt

tt d

2ji t

ejj

t e 1 10

t

ai dt tt e

Average modified decoherence rate

Reservoir response(memory) function

Phasemodulation

Kofman & Kurizki, Nature 405, 546(2000); PRL 87, 270405 (2001); PRL 93, 130406(2004)Gordon, Erez and Kurizki, J. Phys. B, 40, S75 (2007) [review]

11

2

1 2*

2 110 0 2

2Re ai t tt t

t etR d t ttt

tt d

2 a tR t Fd G

Time-domain

Frequency-domain

2

ejG

1

2

1 10

/

1

2

t t

t i tt

F t

dt t e

System-bath coupling spectrum

Spectral modulation intensity

G()

Ft()

R t

Universal dynamical decoherence control formalism

Kofman & Kurizki, Nature 405, 546(2000); PRL 87, 270405 (2001); PRL 93, 130406(2004)Gordon, Erez and Kurizki, J. Phys. B, 40, S75 (2007) [review]

No modulation (Golden Rule)

0 1

2

a

t

a

t t

F

R G

1 10

t

ai dt tt e

Universal dynamical decoherence control formalismSingle-qubit decoherence control

Decay due to finite-temperature bath couplingProper dephasing

Multi-qudit entanglement preservationImposing DFS by dynamical modulationEntanglement death and resuscitation

Dephasing control during quantum computation

(Gordon & Kurizki, PRL 97, 110503 (2006))

(Gordon & Kurizki, PRA 76, 042310 (2007))

(Gordon et al. J. Phys. B, 40, S75 (2007))

R R

( )2 T taGt FR d

G()

A B

11

22

(Gordon, unpublished)

Brief overview of Calculus of Variations

Want to minimize the functional: 0

, ' , , 'T

y y F t y y dtF

0

, ' , , 'T

y y K t y y dt E KWith the constraint:

The procedure:

' '

F F K K

y t y y t y

1. Solve Euler-Lagrange equation

Get solution: ;y t 0 00 ; ' 0 'y y y y

;y t E

; , ' ;y t y t E K

E

2. Insert the solution to the constraint:

Get

3. Get solution as a function of the constraint:

Analytical derivation of optimal modulation

Want to minimize the average modified decoherence rate:

With the energy constraint (a given modulation energy):

11

2

1 2*

2 110 0 2

2Re ai t tT t

t etR d t ttT

tT d

2

0

T

adt Et

1 10

t

ai dt tt e

AC-Stark shift

g|

e|

a

a t

1 10

ti dt t

t e

Resonant field amplitude

g|

e|

t

r t

(Gordon et al. J. Phys. B, 40, S75 (2007))

Analytical derivation of optimal modulation

Want to minimize the average modified decoherence rate:

With the energy constraint (a given modulation energy):

11

2

1 2*

2 110 0 2

2Re ai t tT t

t etR d t ttT

tT d

2

0

T

adt Et

, 0'' tt t Z

1 110 1

1, sin

Tt tt dt t tt t

Tt Z

Euler-Lagrange equation for optimal modulation

0 ' 00

0i tt t e Use notation:

0a

0

t

at d

Analytical derivation of optimal modulation

'' , 0t t t Z

1 1 1 10

1, sin

Tt t dt t t t t t t

T Z

Euler-Lagrange equation for optimal modulation

0 ' 0 0

Using the energy constraint, one can obtain:

2

1 1 10 0

1,

T tE dt dt t t

E Z

12

1 2 2 20 0

,''

,T t

E t tt

dt dt t t

Z

Z

Equation for Optimal Modulation

Numerical resultsCompare optimal modulation to Bang-Bang (BB) control:

Viola & Lloyd PRA 58 2733 (1998)Shiokawa & Lidar PRA 69 030302(R) (2004)Vitali & Tombesi PRA 65 012305 (2001)Agarwal, Scully, Walther PRA 63, 044101 (2001)

a t

0

0

2| | /

t

a a

ti t it

t

t t d

t e d e

F t

Numerical resultsCompare optimal modulation to Bang-Bang (BB) control:

Viola & Lloyd PRA 58 2733 (1998)Shiokawa & Lidar PRA 69 030302(R) (2004)Vitali & Tombesi PRA 65 012305 (2001)Agarwal, Scully, Walther PRA 63, 044101 (2001)

a t

Numerical resultsCompare optimal modulation to Bang-Bang (BB) control:

Viola & Lloyd PRA 58 2733 (1998)Shiokawa & Lidar PRA 69 030302(R) (2004)Vitali & Tombesi PRA 65 012305 (2001)Agarwal, Scully, Walther PRA 63, 044101 (2001)

a t

/

/ 1/ ct

aG

DD condition

1/ ct 2 0aR Fd G

Numerical results1/ noisef

min max1/G

Optimal pulse shape

2 a tR t Fd G

Fidelity Rte

F. T.X

2 a tR t Fd G

Numerical results - multi-peakedG

Optimal pulse shape

• Dynamical decoupling and Bang-Bang modulations are environment-insensitive, i.e. ignore coupling

spectrum

• Optimal modulation “reshapes” (chirps) the pulse to minimize spectral overlap of the system-bath coupling and modulation spectra

• Current results using universal dynamical decoherence control are also applicable to decay and proper-dephasing, at finite- temperatures

• Extensions to multi-partite deocherence and entanglement optimal control underway…

“Know thy enemy” Thank you !!!